Epistemic extensions of combined classical and intuitionistic propositional logic

Logic L was introduced by Lewitzka [7] as a modal system that combines intuitionistic and classical logic: L is a conservative extension of CPC and it contains a copy of IPC via the embedding varphimapstosquarevarphi. In this article, we consider L3, i.e. L augmented with S3 modal axioms, define basic epistemic extensions and prove completeness w.r.t. algebraic semantics. The resulting logics combine classical knowledge and belief with intuitionistic truth. Some epistemic laws of Intuitionistic Epistemic Logic studied by Artemov and Protopopescu [1] are reflected by classical modal principles. In particular, the implications "intuitionistic truth Rightarrow knowledge Rightarrow classical truth" are represented by the theorems squarevarphirightarrow Kvarphi and Kvarphirightarrowvarphi of our logic EL3, where we are dealing with classical instead of intuitionistic knowledge. Finally, we show that a modification of our semantics yields algebraic models for the systems of Intuitionistic Epistemic Logic introduced in [1].